Enhancement of sinusoids in colored noise and the whitening performance of exact least-squares predictors

The extraction of sinusoids in white noise using least squares predictors has attracted a lot of attention in the past, mainiy in the context o€ the adaptive line enhancer (ALE). However, very few results exist for the practicd colored noise case or for the whitening performance of the predictors, We use a matrix formulation to derive the optimal least squares coefficients and frequency response of the D-step predictor or ALE for sinusoids (red or complex) in additive colared noise, Several cases are considered, and in particular, new formulas for the amplitude gain are obtained. In low-pass background noise, the amplitude gain of the sinusoids becomes essentially a monotonically increasing function of their frequency, and a decreasing function for high-pass noise. For the whitening application, signal-to-noise ratio {SNR) bounds of the output are derived when the input is a white signal plus a sinusoidal interference. We also give a state-space model and stochastic interpretations of our analysis of the D-step predictor, providing connections to other related areas. To enable filtering of nonstationary complex inputs, as well as multichannel and multi-experiment data, a complex vector version of the ladder algorithm is presented that can be used to implement the ALE, noise cancdling, and noise inversion for narrow-band interference rejection.